We discuss here a large class of big data problems where MapReduce can't be used - not in a straightforward way at least - and we propose a rather simple analytic, statistical solution.
MapReduce is a technique that splits big data sets into many smaller ones, process each small data set separately (but simultaneously) on different servers or computers, then gather and aggregate the results of all the sub-processes to produce the final answer. Such a distributed architecture allows you to process big data sets 1,000 times faster than traditional (non-distributed) designs, if you use 1,000 servers and split the main process into 1,000 sub-processes.
MapReduce works very well in contexts where variables or observations are processed one by one. For instance, you analyze 1 terabyte of text data, and you want to compute the frequencies of all keywords found in your data. You can divide the 1 terabyte into 1,000 data sets, each 1 gigabyte. Now you produce 1,000 keyword frequency tables (one for each subset) and aggregate them to produce a final table.
However, when you need to process variables or data sets jointly, that is 2 by 2 or or 3 by 3, MapReduce offers no benefit over non-distributed architectures. One must come with a more sophisticated solution.